Allow me to explain why when I say
"I can't prove this to you or anyone else. But by having my thoughts I know that I exist."
It is different than when religious folks say the same thing.
The conclusion that I draw (that I exist) comes from direct observations that I cannot share with anyone else (that I think). Others can have the same observations of themselves, but not of me because when I assert that I think, it could actually be them thinking in their mind.
Anyway, the key is that the observation I draw my conclusions from is not something that I can share or prove to others.
Religious people also claim that they have an observation that they cannot share with anyone else. They claim that they have experienced God, and that that experience is not provable because one must experience it for themselves. On the surface, this looks to be quite similar to my situation but it is not.
I experience thought, and I define myself to be the thinker of that thought.
Religious people experience something, and they define that something to be God.
Parallel so far, and completely acceptable logically.
However, I cannot say ANYTHING further than that I am the thinker of my thoughts. NOTHING further can be concluded. I cannot differentiate any experience I have from my own thoughts or some external force because there is no logical means to do so.
Religious people can also (logically) only prove that they are the thinker of their thoughts the perceiver of their perceptions. But they cannot differentiate between what is in their mind and what is external (just as I cannot). So to take things a step further and say that this particular thing is in my head, and this particular thing is from something ELSE which I call God is not logically sound.
But lets take it a step further. Lets say (beyond that with logic permits) that you could SOMEHOW figure out that part of your perceptions are created external to yourself by what you call God. You still have no means of identifying this thing. You cannot say, for example, that you know it is the God of the new testament. You cant say that it is the devil. You cant say that it is anything that you think might exist outside your own thoughts. So even if you could figure out that it isnt you who is imagining the whole thing you still couldnt figure out what it is.
I cannot prove that I exist to YOU because you cannot see the evidence.
You cannot logically prove that God exists to YOURSELF even though you think you have evidence.
There is a difference.